persistent homology and the stability theorempersistent homology and the stabilitytheorem ulrich...

Persistent Homology

and the

Stability Theorem

Ulrich Bauer

February 19, 2018

TAGS – Linking Topology to Algebraic Geometry and Statistics

http://ulrich-bauer.org/persistence-talk-leipzig.pdf 1 / 31

What is persistent homology?

0.1 0.2 0.4 0.8δ

Persistent homology is the homology of a Bltration

A Bltration is a certain diagram K ∶ R→ Top R is the poset category of (R, ≤) A topological space Kt for each t ∈ R An inclusion map Ks Kt for each s ≤ t ∈ R

Consider homology with coe?cients in a Beld (often Z2)

H∗ ∶ Top→ Vect Persistent homology is a diagram M ∶ R→ Vect

(persistence module)

H∗ ∶ Top→ Vect

Persistent homology is a diagram M ∶ R→ Vect(persistence module)

Homology inference

Given: Bnite sample P ⊂ Ω of unknown shape Ω ⊂ Rd

Problem (Homology inference)

Determine the homologyH∗(Ω).

Problem (Homological reconstruction)

Construct a shape X with H∗(X) ≅ H∗(Ω).

approximate the shape by a thickening Bδ(P) covering Ω

Requires strong assumptions:

Homology inference

approximate the shape by a thickening Bδ(P) covering ΩRequires strong assumptions:

Homology reconstruction by thickening

Theorem (Niyogi, Smale,Weinberger 2006)

Let M be a submanifold of Rd. Let P ⊂ M, δ > 0 be such that

Bδ(P) coversΩ, and

δ <√3/20 reach(M).

Then H∗(M) ≅ H∗(B2δ(P)).

δ <√3/20 reach(M).

0.2 0.4 0.6 0.8 1.0 1.2

δ <√3/20 reach(M).

0.2 0.4 0.6 0.8 1.0 1.2

δ <√3/20 reach(M).

0.2 0.4 0.6 0.8 1.0 1.2

δ <√3/20 reach(M).

0.2 0.4 0.6 0.8 1.0 1.2

δ <√3/20 reach(M).

0.2 0.4 0.6 0.8 1.0 1.2

δ <√3/20 reach(M).

0.2 0.4 0.6 0.8 1.0 1.2

δ <√3/20 reach(M).

0.2 0.4 0.6 0.8 1.0 1.2

δ <√3/20 reach(M).

0.2 0.4 0.6 0.8 1.0 1.2

δ <√3/20 reach(M).

0.2 0.4 0.6 0.8 1.0 1.2

δ <√3/20 reach(M).

0.2 0.4 0.6 0.8 1.0 1.2

δ <√3/20 reach(M).

0.2 0.4 0.6 0.8 1.0 1.2

δ <√3/20 reach(M).

0.2 0.4 0.6 0.8 1.0 1.2

δ <√3/20 reach(M).

0.2 0.4 0.6 0.8 1.0 1.2

δ <√3/20 reach(M).

0.2 0.4 0.6 0.8 1.0 1.2

δ <√3/20 reach(M).

0.2 0.4 0.6 0.8 1.0 1.2

δ <√3/20 reach(M).

0.2 0.4 0.6 0.8 1.0 1.2

δ <√3/20 reach(M).

0.2 0.4 0.6 0.8 1.0 1.2

δ <√3/20 reach(M).

0.2 0.4 0.6 0.8 1.0 1.2

Homology inference using persistence

Theorem (Cohen-Steiner, Edelsbrunner, Harer 2005)

LetΩ ⊂ Rd. Let P ⊂ Ω, δ > 0 be such that

the inclusionsΩ Bδ(Ω) B2δ(Ω) preserve homology.

Then H∗(Ω) ≅ imH∗(Bδ(P) B2δ(P)).

0.2 0.4 0.6 0.8 1.0 1.2

Homological realization

This motivates the homological realization problem:

Problem

Given a simplicial pair L ⊆ K, Bnd X with L ⊆ X ⊆ K such that

H∗(X) = imH∗(L K).

Theorem (Attali, B, Devillers, Glisse, Lieutier 2013)

The homological realization problem isNP-hard, even inR3.

Problem

H∗(X) = imH∗(L K).

Problem

H∗(X) = imH∗(L K).

Problem

H∗(X) = imH∗(L K).

This is not always possible:

L X? K

Problem

H∗(X) = imH∗(L K).

L X? K

Problem

H∗(X) = imH∗(L K).

L X? K

Problem

H∗(X) = imH∗(L K).

L X? K

Problem

H∗(X) = imH∗(L K).

L X? K

The homological realization problem isNP-hard, even inR3.http://ulrich-bauer.org/persistence-talk-leipzig.pdf 8 / 31

Stability

Persistence and stability: the big picture

Data point cloud P ⊂ Rd

Geometry function f ∶ Rd → R

Topology topological spaces (Bltration) K ∶ R→ Top

Algebra vector spaces (persistence module) M ∶ R→ Vect

Combinatorics intervals (persistence barcode) B ∶ R→Mch

distance

sublevel sets

distance

sublevel sets

homology

distance

sublevel sets

homology

structure theorem

Stability of persistence barcodes for functions

Let ∥f − g∥∞ = δ. Then there exists amatching between the

intervals of the persistence barcodes of f and g such that

matched intervals have endpointswithin distance ≤ δ, and

unmatched intervals have length ≤ 2δ.

Interleavings

Let δ = ∥f − g∥∞. Write Ft = f −1(−∞, t] and Gt = g−1(−∞, t].

Then the sublevel set Bltrations F,G ∶ R→ Top are

δ-interleaved:

Ft Ft+δ Ft+2δ Ft+3δ

Gt Gt+δ Gt+2δ Gt+3δ

Applying homology (functor) preserves commutativity

persistent homology of f , g yields

δ-interleaved persistence modules R→ Vect

Interleavings

δ-interleaved:

Interleavings

δ-interleaved:

Geometric interleavings

Persistence modules

A persistence module M is a diagram (functor) R→ Vect:

a vector space Mt for each t ∈ R a linear map mt

s ∶ Ms →Mt for each s ≤ t (transition maps)

respecting identity: mtt = idMt

and composition: mts ms

r = mtr

If each dimMt <∞, we say M is pointwise Bnite dimensional.

Amorphism f ∶ M → N is a natural transformation:

a linear map ft ∶ Mt → Nt for each t ∈ R morphism and transition maps commute:

Persistence modules

A persistence module M is a diagram (functor) R→ Vect: a vector space Mt for each t ∈ R

a linear map mts ∶ Ms →Mt for each s ≤ t (transition maps)

r = mtr

Persistence modules

A persistence module M is a diagram (functor) R→ Vect: a vector space Mt for each t ∈ R a linear map mt

r = mtr

Persistence modules

r = mtr

Persistence modules

r = mtr

Persistence modules

r = mtr

a linear map ft ∶ Mt → Nt for each t ∈ R

morphism and transition maps commute:

Persistence modules

r = mtr

Interval PersistenceModules

Let K be a Beld. For an arbitrary interval I ⊆ R,

deBne the interval persistence module K(I) by

K(I)t =

⎧⎪⎪⎨⎪⎪⎩

K if t ∈ I,0 otherwise,

with transition maps of maximal rank.

Schematic example:

0 0 K K K 0 0

Barcodes: the structure of persistence modules

Theorem (Krull–Schmidt; Crawley-Boewey 2015)

Let M be a pointwise Bnite-dimensional persistence module.

Then M is interval-decomposable:

there exists a unique collection of intervals B(M) such that

M ≅ ⊕I∈B(M)

B(M) is called the barcode of M.B(M)

The decomposition itself is not unique.

This is why we use homology with coe?cients in a Beld.

M ≅ ⊕I∈B(M)

Algebraic stability of persistence barcodes

Theorem (Chazal et al. 2009, 2012; B, Lesnick 2015)

If two persistence modules are δ-interleaved,

then there exists a δ-matching of their barcodes:

matched intervals have endpoints within distance ≤ δ,

Mt Mt+δ Mt+2δ

Nt Nt+δ Nt+2δ

Barcodes as diagrams

The matching category

Amatching σ ∶ S↛ T is a bijection S′ → T′, where S′ ⊆ S, T′ ⊆ T.

Composition of matchings σ ∶ S↛ T and τ ∶ T ↛ U :

Matchings form a categoryMch objects: sets

morphisms: matchings

Barcodes as matching diagrams

We can regard a barcode B as a functor R→Mch:

For each real number t, let Bt be the set of intervals of Bthat contain t, and

for each s ≤ t, deBne the matching Bs ↛ Bt

to be the identity on Bs ∩ Bt .

0.1 0.2 0.4 0.8∆

0.1 0.2 0.4 0.8δ

Stability via functoriality?

Ft Ft+2δ

Gt+δ Gt+3δ

H∗(Ft) H∗(Ft+2δ)

H∗(Gt+δ) H∗(Gt+3δ)

B(H∗(Ft)) B(H∗(Ft+2δ))

B(H∗(Gt+δ)) B(H∗(Gt+3δ))

B(H∗(Ft)) B(H∗(Ft+2δ))

B(H∗(Gt+δ)) B(H∗(Gt+3δ))

Non-functoriality of the persistence barcode

Theorem (B, Lesnick 2015)

There exists no functorVectR →MchRsending each persistence

module to its barcode.

Proposition

There exists no functorVect→Mch sending each vector space of

dimension d to a set of cardinality d.

Non-functoriality of the persistence barcode

Theorem (B, Lesnick 2015)

There exists no functorVectR →MchRsending each persistence

module to its barcode.

Proposition

There exists no functorVect→Mch sending each vector space of

dimension d to a set of cardinality d.

Induced barcode matchings

Structure of persistence submodules / quotients

Proposition

Let f ∶ M → N be amonomorphism of persistence modules:

each ft ∶ Mt → Nt is injective.

Then f induces an injective map B(M)→ B(N)

mapping each I ∈ B(M) to some J ∈ B(N)

with larger or same left and same right endpoint.

Dually for epimorphisms (left and right exchanged).

Induced matchings

For a general morphism f ∶ M → N of persistence modules:

consider epi-mono factorization

M ↠ im f N .

im f N induces injection B(im f ) B(N)

M ↠ im f induces injection B(im f ) B(M)

compose to amatching B(M)↛ B(N):

B(im f )

Induced matchings

For a general morphism f ∶ M → N of persistence modules:

consider epi-mono factorization

M ↠ im f N .

im f N induces injection B(im f ) B(N)

M ↠ im f induces injection B(im f ) B(M)

compose to amatching B(M)↛ B(N):

B(im f )

Stability from interleavings

Consider interleaving ft ∶ Mt → Nt+δ , gt ∶ Nt →Mt+δ (∀t ∈ R):

Nt−δ Nt+δ

ftgt−δ

nt−δ ,t+δ

imnt−δ,t+δ im ft Nt+δ

Mt ↠ im ft

Nt−δ Nt+δ

ftgt−δ

nt−δ ,t+δ

Mt ↠ im ft

Nt−δ Nt+δ

ftgt−δ

nt−δ ,t+δ

Mt ↠ im ft

Nt−δ Nt+δ

ftgt−δ

nt−δ ,t+δ

Mt ↠ im ft

B(N•+δ)B(N)

Nt−δ Nt+δ

ftgt−δ

nt−δ ,t+δ

Mt ↠ im ft

B(im f )B(N•+δ)

Nt−δ Nt+δ

ftgt−δ

nt−δ ,t+δ

Mt ↠ im ft

B(im f )B(im n•–δ,•+δ)

B(N•+δ)B(N)

Nt−δ Nt+δ

ftgt−δ

nt−δ ,t+δ

Mt ↠ im ft

B(N•+δ)B(N)

Nt−δ Nt+δ

ftgt−δ

nt−δ ,t+δ

Mt ↠ im ft

B(N•+δ)B(N)

Nt−δ Nt+δ

ftgt−δ

nt−δ ,t+δ

Mt ↠ im ft

B(N•+δ)B(N)

Sending persistence into

Hilbert space

Extending the TDA pipeline

Mapping barcodes into a Hilbert space?

desirable for (kernel-based) machine learning methods

and statistics

stability (Lipschitz continuity): important for reliable

predictions

inverse stability (bi-Lipschitz): avoid loss of information

Existing kernels for persistence diagrams:

stability bounds only for 1−Wasserstein distance

Lipschitz constants depend on bound on number and

range of bars

no bi-Lipschitz bounds known

Can we hope for something better?

and statistics

predictions

range of bars

and statistics

predictions

range of bars

No bi-Lipschitz feature maps for persistence

Theorem (B, Carrière 2018)

There is no bi-Lipschitzmap from the persistence diagrams

(with the interleaving or any p−Wasserstein distance)

into any Bnite-dimensionalHilbert space,

evenwhen restricting to bounded range or number of bars.

If therewas such a bi-Lipschitzmap into someHilbert space,

the ratio of the Lipschitz constantswould have to go to∞

togetherwith the bounds on number or range of bars.

No bi-Lipschitz feature maps for persistence

There is no bi-Lipschitzmap from the persistence diagrams

(with the interleaving or any p−Wasserstein distance)

into any Bnite-dimensionalHilbert space,

evenwhen restricting to bounded range or number of bars.

If therewas such a bi-Lipschitzmap into someHilbert space,

the ratio of the Lipschitz constantswould have to go to∞

togetherwith the bounds on number or range of bars.

History

When was persistent homology invented?

[Edelsbrunner/Letscher/Zomorodian 2000]

[Robbins 1999]

[Frosini 1990]

[Leray 1946]?

[Robbins 1999]

[Frosini 1990]

[Leray 1946]?

[Robbins 1999]

[Frosini 1990]

[Leray 1946]?

[Robbins 1999]

[Frosini 1990]

[Leray 1946]?

When was persistent homology invented Brst?

When was persistent homology invented Brst?Web Images More… Sign in

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JL Kelley, E Pitcher Annals of Mathematics, 1947 JSTOR

The developments of this paper stem from the attempts of one of the authors to deduce relations between homology groups of a complex and homology groups of a complex which is its image under a simplicial map. Certain relations were deduced (see [EP 1] and [EP 2] ...

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American Mathematical Society. Thus Morse grew to maturity just at the time when the subject of Analysis Situs was being shaped by such masters2 as Poincaré, Veblen, LEJ Brouwer, GD Birkhoff, Lefschetz and Alexander, and it was Morse's genius and destiny to ...

Marston Morse and his mathematical works

M Morse, CB Tompkins Duke Math. J, 1941 projecteuclid.org

1. Introduction. We are concerned with extending the calculus of variations in the large to multiple integrals. Theproblem of the existenceof minimal surfaces of unstable type contains many of the typical difficulties, especially those ofa topological nature. Having studied this ...

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Unstable minimal surfaces of higher topological structure

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The goal of this thesis is to bring together two different theories about critical points of a scalar function and their relation to topology: Discrete Morse theory and Persistent homology. While the goals and fundamental techniques are different, there are certain ...

[PDF] Persistence in discrete Morse theory

MR Hestenes American Journal of Mathematics, 1945 JSTOR

1. Introduction. Let f (x)=(. z,**, xn) be a function defined on a neighborhood of a closed set S in euclidean nspace. Suppose that the points of S at which f, 0 0 are interior to S and that the determinant I is different from zero at these points. A point at which f, 0 is called a ...

A theory of critical points

DG Bourgin Annali di Matematica Pura ed Applicata, 1955 Springer

Summary The properties of direct limits of Cech homology groups for a non cofinal collection of compact pairs are explicitly stated with a view towards applications. The usual axioms including full excision are satisfied under obvious restrictions. Some of the Morse rank and ...

Arrays of compact pairs

M Morse, M Heins Annals of Mathematics, 1946 JSTOR

Paper II of this series* starts with a theorem relating the number of logarithmic poles and essential critical points of a harmonic or pseudoharmonic function F (x, y). Cf.? 2. II. The function F was defined on the closure G of a finite region bounded by v Jordan curves.

Topological Methods in the Theory of Functions of a Single Complex Variable: III. CasualIsomorphisms in the Theory of PseudoHarmonic Functions

D Pope Pacific J. Math, 1962 msp.org

Introduction^ A basic feature of most of the methods used for the numerical calculation of a variational problem is the reduction of the infinite dimensional problem to a finite dimensional problem by some kind of approximation. One of the most natural ...

[PDF] On the approximation of function spaces in the calculus of variations

M Ozawa Kodai Mathematical Seminar Reports, 1958 jlc.jst.go.jp

Let W and W be two closed Riemann surfaces of the same genus g Ξ> 2. Let § be a preassigned homotopy classs of topological mappings T: W+ Wf and ξ)(q) be a subclass of ξ), each member of which carries a given fixed point J) o on W to a point q on W. Let T (qp) ...

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On extremal quasiconformal mappings

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ed.ac.uk [PDF]

ams.org [PDF]

psu.edu [PDF]

msp.org [PDF]

projecteuclid.org [PDF]

Morse’s functional topology

Key aspects:

early precursor of persistence and spectral sequences

uses Vietoris homology with Beld coe?cients

applies to a broad class of functions on metric spaces

(not necessarily continuous)

inclusions of sublevel sets have Bnite rank homology

(q-tame persistent homology)

focus on controlled behavior in pathological cases

Motivation and application: minimal surfaces

(from Dierkes et al.: Minimal Surfaces, Springer 2010)

Existence of unstable minimal surfaces

Using persistent homology:

Number of є-persistent critical points (minimal surfaces)

is Bnite for any є > 0 Morse inequalities for є-persistent critical points

Theorem (Morse, Tompkins 1939)

There is aC1 curve bounding an unstable minimal surface

(an index 1 critical point of the area functional).

Thanks for your attention!

persistent homology and the stability theorempersistent homology and the stabilitytheorem ulrich...

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